Real Analysis Proofs I am taking a Real Analysis class using the textbook Analysis with an Introduction to Proofs, $5^{th}$ Ed. by Steven Lay. So far I am not understanding the proofs at all. Does anyone know of any good resources that can help me understand the proofs or any key techniques that aides in doing them?
An example would be 
Let A and B be subsets of a universal set U. Simplify the expression
(A v B) ^ [ A v (U\B)]
 A: You may find Chardrand, Polimeni & Zhang's Mathematical Proofs: A Transition To Advanced Mathematics useful. They teach us the intro class from this book in my university, and it has a whole chapter on proofs in  advanced calculus (or introductory analysis, if you like). I have not read that chapter thoroughly (we don't learn about analysis in the intro class), but I have taken a look at it, and it has all these "proof strategy" parts before actual proofs, and the book in general does not skip any steps (i.e., no "proof is easy"'s :) ). Another suggestion would be Fitzpatrick's Advanced Calculus. Again, this book is very thorough with the proofs. There are lots of exercises in both of the books as well.
As for my humble suggestion on how to understand the proofs, I find it quite beneficial to first convincing oneself that the theorem must be the case by coming up with different examples. Modify the hypotheses and see where the theorem really breaks down. This is a slow process at first but the examples/counterexamples start to come to you faster (in general) as you continue. Finally, hang in there :) .
As a final remark, I find the following (trivial) lemma quite useful in proving results related to sequences: A sequence converges to $x$ iff for any subsequence of it there is a further subsequence that converges to $x$. This may count as a useful argument.
A: When it comes down to it, learning how to read and write proofs is a matter of practice. If you keep working and quizzing yourself, you'll get the hang of it eventually. As for tips, here are a few I typically give to students I've TA'd for:

*

*Take careful note of your assumptions, and make sure you use all of them in your proof. (If you're trying to prove "Suppose $X$ and $Y$. Then $Z$." and your proof relies only on $Y$, your proof is probably wrong.

*Review your definitions. If you want to prove "all multiples of $4$ are even", a good way to start is by translating "multiple of $4$" and "even" into their definitions, and trying to connect the two. In this case, you might say "even" means a multiple of $2$, "multiples" are products of a number and an integer, so if $x$ is a multiple of $4$, $x = 4k = 2(2k)$, and is therefore even. (this example might seem a little silly, but it really does help when you're working with more complex concepts, like continuity or boudedness).

*Try different modes of attack. It might be easier to prove the contrapositive of the statement, or do a proof by contradiction instead.

*Try to visualize or sketch the problem. This is not always going to be helpful (like in proving facts about numbers), but for proofs involving graphs or sets it can help you gain an intuition for what you're doing.

*Test the proof with concrete examples. This can be typically enlightening, since it often shows why we need each assumption in a given step.

You can find lots of examples of proofs online, and I'm sure if you look around a bit you can find much better advice/tips on how to understand proofs already on stack exchange. My biggest tip though is to work with your classmates and professor. They're actually there--they can explain things to you (they may have even had the same confusion you've had about a particular proof). Good luck!
A: I used this book as well for my Elementary Real Analysis course and found it to be quite good. I recommend reading through the examples several times to fully understand them. In the beginning sections, there are also some "fill in the blank" proofs to get you started. One thing I found out about while taking (Elementary) Real analysis, is that it is perfectly normal to be "stumped" for a while on a question. Some of the questions asked simply don't have very straight forward or intuitive answers at first glance. For an example, use of triangle inequality to prove that Convergent $\Leftrightarrow$ Cauchy. But, the more questions you do, the easier it is to see such tricks, and at least know how to start. If you can edit your post to include a question you are having trouble with, I would be more than happy to try and help you with it.
